* Appendix A: Spherical Tensor Formalism
The geometrical aspects of the quantum-mechanical properties of molecular systems are described in terms of quantities familiar from angular momentum theory and the reader is referred to the text books existing on this subject. 1- 3 In this appendix we collect together the quantities and relations most fre• quently used in calculations concerning the properties of solid hydrogen and other systems of interacting molecules.
The spherical harmonics Y,m(O,
(1m I I'm') == SOh S: Y;';" Y,'m,sine de d 277 278 Appendix A • Spherical Tensor Formalism F or this choice of phases Y'tr,(8,c/J) = (-lrr;m(8,c/J) (A.5) where m == -m. Many expressions are simplified by using the Racah spherical harmonics defined as (A.6) which satisfy (A.7) where PI is the Legendre polynomial of degree I. For I = 0, ... , 4 and m > 0 these functions are given in Table A.1, and for m > 0 can be obtained from eg. (A.5). Hence explicit minus signs in the expressions for the elm and r;m occur only for positive odd values of m. The transformation law of the spherical harmonics under rotations of the coordinate frame is (A.S) m where 8,c/J and 8',c/J' are the polar angles of a given direction (unit vector) in the old and the new frame, respectively, and ~, /3, " are the Euler angles of the rotation carrying the old into the new frame. This rotation is effected by Table A.1 The Racah spherical harmonics up to I = 4. Coo = 1 C IO = cosO C 20 = t(3cos28 - 1) C 30 = lcos8(icos18 - 1) C40 = i(¥cos4 0 - lOcos28 + 1) C ll = -rtll/2sin!ie'4> C 21 = - (~)1!2sin8cosOe'4> C 31 = -i,/3sin8(5cos28 - Ije'4> C41 = -i,JSsin8cos8(1cos28 - l)e,4> C22 = t(~)1'2sin28e2i,p C 31 = HY/ 2sin18cos8e2',p C 41 = i(W 2 sin28(7cos28 - l)e2i 4> C 33 = -i.j5sin38e 3,,p 3 C43 = -iJSsin38cos!ie ',p C 44 = ~"isin 40e4,,p A.1 • Spherical Harmonics 279 first rotating the old frame x,y,.: into X 1,Y1,':1 = .: by a rotation through an angle oc around the z axis, then through an angle [3 around the )' 1 axis to produce a frame XhV2 = )'1' ':2 = .:' and finally through an angle ~' around the z' axis to produce the new frame x',y',.:'. The same transformation is obtained by rotations around the original axes, first through ~' around .:, then through [3 around y, and finally through :x around.: again. The rotatioll operator in the Hilbert space of a quantum-mechanical system with total angular momentum J. corresponding to the rotation R = (:x.[3.~') is and the rotatioll matrices are defined as D!",,(R) = (jmiD(R)ijll> = e-im~d~n([3)e-in;' (A.10) where d;m,(/3) = (jmie-i/lJ'ijll> (A.11) The rotation matrices are unitary (A.12) LD~n(R)*D!nll,(R) = b'III' m and satisfy the symmetry relations D~n(R)* = (-l)m-nD~n(R) = D~m(R-1) (A.13) where R - 1 = (-~' + n, [3, -:x + n) is the rotation inverse to R, and the orthonormality relations (" (2" (2,,"". 8n 2 Jo Jo Jo D:"n(:x[3~')* D:"·II·(:x[3·,')sin[3 d[3 d:x d~' = 2j + 1 bjj.bmm ·b,I/J' (A.14) For integer j, j = I, and Il = 0 the rotation matrices reduce to spherical harmonics (A.15) and eq. (A.8) then reduces to the spherical harmollics additioll theorem, (A.16) III where 8 is the angle between the directions specified by 8b ¢1 and 82'¢2' The matrices (A.11) have the symmetry properties d!"n([3) = (-l)m-lId~m([3) = dkm([3) (A.17) and forj = 1,2 are given in Table A.2. 280 Appendix A • Spherical Tensor Formalism Table A.2 The reduced rotation matrices d~" for j = 1,2. dl[ = cos21fJ di o = -(WI2sinfJ dbo = cosfJ dlr = sin21fJ d~2 = cos41fJ dL = 10 + cosfJ)(2cosfJ - 1) dL = -1sinfJ(1 + cosfJ) dio = _(~f2sinfJcosfJ d~o = (~)[ 2sin2fJ dh = 10 - cosfJ)(2cosfJ + 1) dh = -1sinfJ(1 - cosfJ) d~2 = sin41fJ A.2 Clebsch-Gordan Coefficients The vector-addition or Clebsch-Gordan coeffiCients are the coefficients C(i! ;2i~;m,m,m~) = <;, ;,m,m, 1;1;' ;m> (A.18) in the unitary transformation Ijd2jm) = I Ijd2mlm2)Ud2mlm2 Ulj2jm) (A.19) corresponding to the addition, J = J 1 + J 2, of the angular momenta J 1 and J 2' The unitarity implies the relations I C(jljzj;m 1m2m )C(jljzj';m1m 2m') = (jjj,(jmm' (A.20) I C(jljzj;mlmzm)C(jdzj;m~m2m) = (jm[m\(jm2m2 jm and the relation J = J I + J 2 implies C(jljzj3;m 1m Zm 3) = 0, unless m3 = m1 + m2, and l'l(jlj2j3) (A.21) where the triangle condition l'l(jlj2h) means that j3 is one of the numbers (A.22) The phases are chosen in such a way that all the coefficients (A.18) are real and (A.23) where C(jlj2j3;m 1mZ ) == C(jdzj3;m 1,mZ,ml + mz) (A.24) We note the further properties C(jjOj3;m I 0) = (jith (A.25) Sec. A.2 • Clebsch-Gordan Coefficients 281 and C(jjj2j3;OO) = O. unless 2J == jl + j2 + h is even. (A.26) For J integer, i.e.,jl + j2 + j3 even, we have J . J! C(jlhi3;OO) = (-1) - J3 (J _ JJ!-(J---j-z)-!(-J---j-'3-)! x i(2i3 + 1)(j1 + j2 - j3)!(jl - j2 + h)!( -j~+ j2 + j3)!jl 2 L (2J+l)! (A.27) where 2J = jl + jz + j3' The symmetry properties of the Clebsch-Gordan coefficients are most easily expressed in terms of the Wigner 3-j symbol defined by which is invariant under cyclic permutations ofthe columns, and is multiplied by a phase factor under noncyclic ones = = (_ l)a + b + c (a b c) (b c a) (h a '/c) . etc. (A.29) 'X f3~' f3"/ 'X f3 'X and (~ ~ ~) = (_l)a+b+C(a b c) (A.30) 'X f3 7 'X f3 7 gIvmg C(jlj2h;11l1111 2111 3) = (_I)h + h-hC(jlj2j3;ifllifl2ifz 3) 2' 1)1/2 = (_I)h+m2 ( J.3 + C('hhh .. ;11121113-111 1 ) 211+ 1 2' + 1)12 = (_ l)h +",2 ( -'1-=.3'---_ C('"hhh ;1113- 111 2ml-) 211+ 1 = (_I)h -m, (-,21-=>_+_11)1/2 C(j3jlj2;111 3ifz I 111 2) 212 + (A.31) The numerical values of some often occurring Clebsch-Gordan coefficients can be obtained from Table A.3 and eg. (A.31). More extensive tables are available.4 . 5 282 Appendix A • Spherical Tensor Formalism Table A.3 Numerical values of frequently occurring Clebsch-Gordan coefficients; C = C(/t1zl;mtmzm). 1,/ 2 1 1n 11112 m C 1,lzl m}m2m C 1 1 1 000 0 223 1 1 2 0 1 1 1 1 0 1 (W'z 223 2 1 3 (WIZ 1 12 000 (W z 224 000 3(is)12 1 1 2 1 10 (i)'z 224 iTO 2(is)1/2 1 1 2 101 (l:)' Z 224 210 (10)12 1 12 1 1 2 224 10 1 (W z 224 2 11 (i4)' 2 212 000 0 224 202 (1:,:)' 2 212 o 1 1 _(1)12 224 1 1 2 2(W z 2 I 2 101 (i)12 224 2 I 3 (WO z 212 202 (W 2 224 224 1 2 I 2 1 1 2 -(W 2 3 I 3 000 0 2 I 3 000 (W 2 3 1 3 01 1 -(W 2 2 1 3 110 (!)IZ J ! J ! 0 ! ~!~}1.'2 2 I 3 o I I (W Z 3 1 3 1 1 2 z 2 I 3 101 2(is)1 Z -1(W 3 I 3 202 Ct)t2 2 1 3 2T 1 (is)12 3 1 3 2 1 3 I 2 1 3 I 12 (W 2 -2 3 1 3 303 z 2 1 3 202 (t)1 2 1(W 2 I 3 2 I 3 3 1 4 000 2(W Z 3 I 4 1 10 (1:,:)1 Z 222 000 -(W z 3 1 4 101 1(1;)IZ 222 101 _(i4)12 3 1 4 I I (~)I 2 222 I 12 -(W 2 o 3 1 4 2T 1 1(W Z 222 202 (W 2 3 I 4 202 (W 2 223 000 0 3 1 4 1 1 2 1(1;)12 2 223 1 I 1 (W 3 I 4 312 1(W 2 (io)12 223 220 3 1 4 303 (W 2 /2 223 1 0 1 (W 314 2 1 3 1(W 2 (1o)1!2 223 2 I I 3 I 4 3 14 I 223 202 (W/2 The Clebsch-Gordan coefficients also occur in the decomposition rule for the rotation matrices, D!,:l n l(R)D!;2n2(R) = L C(j Ij2j;m 1 m2f1)C(j d2j;n 1 n2v)D{,,(R) jl1\' L C(j d2j;ml m2)C(j Ij 2j ;nln2)D~1 +mpi +1I2(R) (A.32) Sec. A.3 • Spherical Components of Tensors 283 which corresponds to the reduction of the product representations of the rotation group into their irreducible parts. For j I = II and j 2 = 12 integer and III = 112 = O. eq. (A.32) reduces to the decompositioll rule for the spherical harmonics, which can be written in the form CI1m1 (G,r/J )CI2m2( (},r/J) = I C(l 1121 ;OOO)C(l 1121 ;mlm11ll )Clm(ll,r/J) (A.33) 1m or (A.34) where II + 12 + I is e\en. A.3 Spherical Components of Tensors The spherical compollents .4" of (/ ['ector A are defIned in terms of the Cartesian components by (A.35) and transform under rotations in the same way as the spherical harmonics of order I = 1. which are given by 1 =+= ---= (.\ ± ir) (A.36) ') '\ - where:: = cosH . .\ = sintlcosr/J ..r = sinOsinr/J. Out of two vectors A and B one can form tensors of rank O. 1. 2 by writing (A.37) mil U sing Table A. I. one easily verifies that I Too = ----=A· B (A.38) ,.,,3 and I Too = -. (3AoBo - A . B). - '\ 6 284 Appendix A • Spherical Tensor Formalism For I = lone obtains (A.40) which is a pseudovector. The irreducible parts of a symmetric, second-rank tensor with Cartesian components Tij = Tji are given by (A.41) and 1 T 20 = J6 (2Tzz - T xx - Tyy), (A.42) T 2 . ± 2 =i(Txx - Tyy ± 2iTxy) as follows from eqs. (A.38, 39) by putting AiBj = BiA j = Tij' The results (A.41), (A.42) can be derived in a more straightforward way by writing the relation (A.35) between the spherical and Cartesian components of a vector in the form (A.43) m where U mi is the unitary matrix x y z o I 0 0 1 1 0 Umi = J2 J2 (A.44) -1 0 J2 J2 Since tensors transform as products of vectors, we get from eq. (A.37) (A.45) mn mn ij which is the same as eqs. (A.41), (A.42). Using eq. (A.20) and the unitarity of the matrix (A.44), we obtain from eq. (A.45) the inverse relation Tij = I (UmJ*(Un)* I C(11l;mn,u)T11l (A.46) mn l!1 With the help of these relations one can transform any vector relation from A.4 • Racah Coefficients 285 C artesian into spherical components and z;ice cersa. For example, from (A.47) where Tij = Tji and B, = Bj is real, we get Am = I UmiT,jBj = I C(llI;m,j1 - I11)TI!l(B 1l - m)* (A.48) ij [tint where (Bn)* = (-1 )nB_1I" The use of spherical components is also very convenient in obtaining derivatives of expressions involving spherical harmonics with the help of the gradient /c)rmula 1 V1J(r)Clm(l},c/» = I f~f(r)C(I,l,I + p;mp)CI+p,m+ll(O,c/» (A.49) p= ± 1 where ? 1 Vo=-;;-, V±l = +~ (c-;;- ± i-;;-c) (A.SO) c:; ~2 ex cy :; = rcosO, x = rsin8cosc/>, y = rsinOsinc/>. and fit (/+1)12(C I) = 21 + 1 cr - ~ (A.S1) I )L2 ( c I + fl_t --- ( 21+1 cr + -r-1) are operators acting on f(r). A.4 Racah Coefficients In the coupling of three angular momenta, Racah coefficients or 6-j symbols appear, which depend on six angular momenta quantum numbers. and which are related to each other by a phase factor, {: ~;} = (-l)a+b+c+dW(abcd;ef) (A.S2) The 6-j symbols should be carefully distinguished from the 3-j symbols which contain three angular momenta and three projection quantum numbers, usually denoted by Latin and Greek letters, respectively, whereas the 6-j symbols contain six angular momenta quantum numbers. The 6-j symbols (A.52) are invariant under an interchange of any two columns and of the 286 Appendix A • Spherical Tensor Formalism upper and lower arguments of any two columns, f { a be} = {b a e} = {a c 'e } , etc. (A.53) dcf cdf db The most important application of these quantities is in the contraction of sums of products of Clebsch-Gordan coefficients. The most symmetrical expression for this contraction is obtained by using 3-j and 6-j symbols L (- 1Y + {J +; (A ~ c ) (B C a ) (C A b) x{J,' a [3 y' [3 y a' }' (i [3' = (_ 1 A + B + C {a be} (a be) (A. 54) ) ABC a' [3' / In terms of Clebsch-Gordan and Racah coefficients, the contraction (A.54) can be written in the form L C(abe;a[3)C(edc;a + [3,";' -a - [3)C(bdf;[3,')' - a - [3) {J = r(2e + 1)(2f + 1ll 1/2 W(abcd;eflC(afc;a,:' - a) (A.55) With the help of eq. (A.20), this can also be written in the form L C(abe;a[3)C(edc;a + [3,~' - a - [3)C(bd/;[3,i' - a - [3)C(afc;a,~' - a) = [(2e + 1)(2f + 1)]1/2W(abed;~n (A.56) or equivalently 'li~./-1)a+{J+7(: ; ~,)(Z ~ :)(~ : ~,)(:, ~,~,) ( - 1)A + B + c, , be} {a (A.57) = 2e + 1 OcfOy'' ABC Extensive tables of the numerical values ofRacah coefficients are available. 5.6 Special cases occur for e = 0 (-1t+c- f W(abed;Of) = [(2a + 1)(2e + 1)]1/2 c5 abc5 cd (A.58) and for e = a + b W(abed'a+ b f)= [ (2a) !(2b) !(a +b+ e+d+ 1)!(a+b+e-d)! , , (2a+2b+1)!(e+d-a-b)!(a+e- f)!(a+f -e)! (a+b+d-e)!(e+ f -a)!(d+ f -b)! J1 /2 x (a+e+f + 1)!(b+d-f)!(b+f -d)!(b+ f +d+ 1)! (A.59) Sec. A.S • Wigner-Eckart Theorem 287 In the coupling cJf four angular momenta the addition may be performed in three different ways and the connections between the resulting three different coupling schemes define the Wigner 9-j symhols, or Fano X• coefficients X(abc,dej,yhi) = j ~ ; ~l ly h d = I(2j + l)W(aidh;jg)W(bfhd;je)W(aibj;jc) (A.60) The 9-j symbols are invariant under an interchange of rows and columns, and are multiplied by ( - 1)k, where k is the sum of the nine arguments, upon an interchange of any two adjacent rows and columns. If one ofthe arguments is zero, the sum in eg. (A.60) reduces to one term a b = -I c- q 6 __ { d e c}r -a-"6 W(uhde;cg) (A.61) () cf gh [(2c + 1)(2y + I)r 2 y h o F or further properties and contraction formulae we refer to the literature. 2.7 A.S Wigner-Eckart Theorem In many calculations concerning interacting molecules the matrix elements of angular functions or operators are required between spherical harmonics or more general angular momentum eigenstates. The techniques available for performing such calculations are based on the Wigner-Eckart theorem and Racah algebra, and we collect here the most often used results derived along these lines. From the decomposition rule (A.33) for the Racah spherical harmonics defined by eq. (A.6) one obtains the following expression for the matrix elements of the Rucuh spherical harmonics, (A.62) where dOl = sinO dO de/>. Note that the order of the arguments in the Clebsch• Gordan coefficients corresponds to that in the matrix element read from righ t to left. 288 Appendix A • Spherical Tensor Formalism The Wigner-Eckart theorem is a generalization of eq. (A.62) and states that the matrix elements of any spherical tensor operator Tim between total angular momentum eigenstates depend in the same way on the projection quantum numbers as in eq. (A.62) The symbol cx represents one or more quantum numbers specifying the part of the state other than the angular part, and has been added in eq. (A.63) to show that the reduced matrix elements depend on all the quantum numbers other than the projection quantum numbers. The reduced matrix elements are often defined as (2jz + l)I/Z times the quantities defined here by adding a factor (2jz + 1) -1/Z in the right hand side of eq. (A.63), and one must always check which definition is being used. With the definition adopted here the sum rule following from eq. (A.63) is (A.65) or As indicated by the double bars, the reduced matrix elements are not matrix elements in the usual quantum mechanical sense. In particular, their behavior under an interchange of the two states, which follows easily from the property (A.67) is given by + l)I/Z The advantage of the definition (A.63) is the simplicity of the rule (A.63) for forming the reduced matrix elements, in particular for the usual case that j = I is integer when eq. (A.63) reduces to A.6 Bipolar Harmonics The arguments of the two functions in the left hand side of the coupling rule (A.34) are the same. If these arguments refer to different degrees of freedom, such as the orientations of two different molecules 1 and 2, the resulting quantity is called a bipolar harmonic (A.70) This quantity transforms irreducibly as a spherical tensor of rank I under simultaneous, identical rotations of the two molecules, generated by the sum of the two angular momentum operators. J = J 1 + J 2' Note that the centers-of-mass of the molecules remain fixed under such rotations, since J does not contain the angular momentum of the relative translational motion of the molecules. If I'Y. 1JIM 1) and I'Y. 2J 2M 2) are angular momentum eigenstates of the two molecules, one can form the eigenstates of J2 and J= I'Y.J 1J 2JM) = L C(JJ2J;MIM2M)I'Y.IJ1MI)I'Y.2J2M2) (A.71) M,A12 and according to the Wigner-Eckart theorem (A.69) the matrix elements oj' the bipolar harmonics between the states (A.71) are then given by <'Y.J J 2J'M'ITl~/z)lexJ IJ 2 JM ) = C(JlJ';MmM') Cj~/2)(ml.m2) = L C(l1/2/;mlmlm)C/,m,(mdC/2mz(ml) (A.75) rntm2 with reduced matrix elements (A.76) 290 Appendix A • Spherical Tensor Formalism A.7 Generalized Equivalent Operators It is useful to introduce a generalization of these formulae, closely related to the concept of equivalent operators. For fixed rx and 1, the 21 + 1 states irxl M) of a system with total angular momentum J form a complete set of states in the subspace :If~J of the Hilbert space :If of the system. Sim• ilarly, the (21 + 1f operators irxl M) F = L irxlM) Clm(txl) = L irxlM)C(llJ;NmM) Clm,(rxl) = L irxlM')'C(llJ;N'm'M') The operators (A.78) are defined in the full Hilbert space :If of the system, but have nonvanishing matrix elements only within :Ifu where they can be represented by polynomial expressions of degree I in the components of the angular momentum operator J, the polarized spherical harmonics Clm(J), or equivalent operators.l,s For example, for I = 1 it is clear that the components 1m of J transform in the same way as the operators Clm(rxl), and according to the Wigner-Eckart theorem the two sets of operators are therefore equivalent within a subspace 1 = const., i.e, (A.81 ) A.7 • Generalized Equivalent Operator 291 The constant of proportionality can be found by calculating one matrix element of the left and right hand side ofeq. (A.81). or from the relations I(-I)mc1m c1", = I. I (- I )mJmJ", = J(J + I) (A.82) m m and is given by (A.83) For 1= 2 a complication arises. In replacing X.r..:: in r 2 C 2m(li.¢) by J".J)".J o' 2 one sees that in r C 21 x .::(x + iy) the quantities J o and J ~ do not commute. The rule for forming the correct equivalent operators8 is to replace products such as JJ + by the symmetrized sum t(JJ ~ + J _Jo ). The I = :2 polarized spherical harmonics are therefore giwn by C20(J) = H3J~ - J(J + 1)]):2 C 2.:cM) = ::n(~)12(JJ± + J±Jo )):2 (A.84) C\.=2(J) = t(W 2Ji):2 where (A.85) as can be verified using eq. (A.27). H is clear that the Clm in general do not commute with each other although the Clm always do. The Clm((1.¢) are functions of the configurational coordinates (I.¢ and in the Schrodinger picture are simply multiplication operators which commute with each other in J(. This implies that I (aiClmi):JM)():JMiClmib) = I (aiClmi):JM)():JA/iClmib) (A.86) ~M ~M where the sums extend over a complete set of states inYf. but the equality (A.S6) is not true term by term nor when summed only over M. Since (aiClm():J)Clm():J)ib) = I (aiClm():J)i):JM)():JMiCI'm,():J)ib) (A.87) .H involves only a sum over 1'vl. one can see why Clm and CI'm' do not commute. The main advantage of using the Clm is that the closure property may be used within the subspace J = const.. as shown by eq. (A.S7). Out of the operators (A.78) for two different or equal subspaces corre• sponding to ):IJ 1 and ):2J 2. one can construct gcncrali.::cd bipolar harmonic Ojlcrutors of the form Cj~,~12J():IJ 1'):2J 2) = I C(l1121:IIJIIIJ211l)C/lm,():IJ 1 )C/2m )):2J 2) (A.S8) /111/112 292 Appendix A • Spherical Tensor Formalism and the matrix elements of these operators are given by formulae identical to eqs. (A.72), (A.76). Introducing the projection operators p,J = ~]xJM> (A.91) 1m lIt, where x = C(J 1k1F1 ;OO)C(1zkzFz ;00)C(F1k'J 1 ;OO)C(Fzk~J z;oo)O(1 1Jzkk') (A.92) and k' z (A.93) Fz References I. M. E. Rose, Elementary Theory of Angular Momenta, J. Wiley and Sons, New York (1957). 2. D. M. Brink and G. R. Satchler, Angular Momentum, Oxford Univ. Press, London (1975). 3. U. Fano and G. Racah, Irreducible Tensorial Sets, Academic Press, New York (1959). 4. Tables ofClebsch-Gordan Coefficients, Institute of Atomic Energy, Academia Sinica, Peking, Science Press (1965). 5. M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Jr., The 3j- and 6j- symbols, Technology Press, MIT, Cambridge, Mass. (1959). 6. A. F. Nikiforov, V. B. Uvarov, and Yu. L. Levitan, Tables of Racah Coefficients, Macmillan, New York (1965). References 293 7. K. Smith and J. W. Stevenson. A Table of Wlgner 9, Coefficients for Integral and Half• Integral Values of the Parameters. Argonne National Laboratory (1957). X. K. W. H. Stevens. Matrix clements and operator equivalents connected with the magnetic properties of rare earth ions. Proc. Phys. Soc. (Lolldoll) A65. 209 215 (1952). 9. S. Luryi. Products of generalized equivalent operators in angular momentum theory. Call. J. Pill'S. 57. 327 329 (1979). * Appendix B: Lattice Sums The sums of inverse powers of the separations over all the sites surrounding a central lattice site in the fcc and hcp lattice are denoted by s" = 2:' (RO!RJ' (B.l) where Ro is the nn separation. The values of some of these sums are given in Table B.1.1 These and other lattice sums are most easily evaluated using the methods of Ref. 2. The number of neighbors in the various shells and their distances from the central lattice site are given in Table B.2. 1 Table B.1 Values of the lattice sums S" in the fcc and hcp lattice. /! fce hep 6 14.45392 14.454g9 8 12.80194 12.80282 10 12.31125 12.31190 12 12.13188 12.13229 14 12.05899 12.05923 16 12.02736 12.02748 18 12.01300 12.01306 295 296 Appendix B • Lattice Sums Table B.2 Squared distances and numbers of neighbors in successive shells in the fcc and hcp lattices, Ref. 1. fcc hcp (R,JRo)2 n, (RJR o)2 n, 1 12 12 2 2 6 2 6 3 3 24 2~ 2 4 4 12 3 18 5 5 24 3~ 12 6 6 8 4 6 7 7 48 5 12 8 8 6 513 12 9 9 36 6 6 10 10 24 6t 6 11 11 24 6~ 12 12 12 24 7 24 13 13 72 7t 6 14 15 48 8t 12 15 16 12 9 12 16 17 48 9~ 24 17 18 30 10 12 18 19 72 lot 12 19 20 24 1~ 2 20 21 48 11 12 References I. T. Kihara and S. Koba, Crystal structure and intermolecular forces of rare gases, J. Ph),s. Soc. (Japan) 7,348-354 (1952). 2. B. R. A. Nyboer and F. W. de Wette, On the calculation of lattice sums, Physica 23,309-321 (1957). * Index Absorption coefficient, 119 per nn pair in linear chain, 230 microwave, 207 per nn pair in Pa3 structure, 235 see a/so Integrated absorption coeffi• see a/so Excess binding energy cient Bipolar harmonics, 289 Accidental symmetry, 56, 90 generalized operator, 291 Adiabatic approximation, 2, 3 matrix elements of, 289 for intermolecular potential, 99 Bloch's theorem, 64, 92 Alternating factor, 97 Bloch waves Amplitude of vihration librational, 244 of harmonic oscillator, 20 orthonormality of, 77 of lattice oscillators, 146, 151 rotational, 92, 96 Anharmonicity constants, 20 vibrational, 63 perturbed, 59 Bogoliubov transformation, 252 Anharmonic lattice effects, 139 Bound complexes Anisotropic Debye model: see Debye the 5, (0),107-109 model the 5,(0) + 5,(0), III Anisotropic interaction of two rotons, 99, 103 in arbitrary frame, 35 of vibron and roton, 106 expansion in spherical harmonics, 33 Broadening of J = 1 impurity in J = 0 solids, 177 of J = I pair levels, 268 single-molecule and coupling terms, 34 of MW absorption lines, 206 see a/so Intermolecular potential Broken symmetry, 87, 229, 251 Anisotropy of lattice vibrations, 162, 177: see a/so Debye model Cancellation effect, 123 Average Schrbdinger equation, 151, 153, cia ratio, 56 156 deviation from close-packed value, 90, Axially symmetric interactions, 212, 216 162, 175 Cell indices, 54 Band origin Cell potential of J = 2 band, 92 in classical crystal, 132-133 of J = I band in solid HD, 101 in quantum crystal, 140-141 of one-libron band, 24 Center-of-charge, 98 of v = 1 band, 63, 73 transformation from center-of-mass to, Binding energy 34,99 of J = 1 pairs, 229, 263 Centrifugal potential, 20, 67 297 298 Index Charge density Debye sphere, 162, 166, 187,222 in J = 0 molecule, 7, 10 Debye temperature, 167, 168 in J = 1 molecule, 8, 10 Decay rates of J = I pairs, 274; see a/so Clebsch-Gordan coefficients, 280 Ortho-para conversion Close-packed structures, 53 Deformation parameters of homogeneous ordered, and disordered, 54 deformation, 156 Clusters of J = 1 impurities, 197,274 Degree of frustration in ordered J = I Coherent states, 148-151 solids, 229, 233; see a/so Frus• of lattice vibrations, 150, 177, 178 trated alignment Cohesive energy of solid, 57 Delocalization EQQ contribution, 84 of bound vibrational state, 69-70 zero-point contribution, 33 effect on Raman cross sections, 76-81 Compressibility, 162 of J = 2 excitations, 107, 114 Condon and Shortley phases, 277 Density-of-states Configurational hindrance, 200 in J = I band, 101 Configuration interaction method, 42 in J = 2 band, 93 Connection between spin and statistics, 23 in vibrational band, 66 Contraction of 3-j symbols, 286 in v = I, J = 2 band, 127 Conversion process: see Ortho-para con- Deuteron nuclei, 26, 266 version Dielectric screening, 110, 174,220,239 Correlation function Diffusion model, 266, 273 of angular momenta, 262, 264 Dipole-dipole interaction, 37, 38, 122 of eiectromagnetlc lIeld, IU Direction cosines, 116, 120 of lattice displacements, 167, 169 averages involving, 121 Correlation time of J = I impurities, 262, Dispersion interaction, 37 264-265 anisotropy factors, 41 Critical ortho-para concentration, 227, 235 long-range anisotropic, 40 Critical pressure, 252 long-range isotropic, 40 Crystal-field interaction Dispersion law equivalent-operator form of, 182 in anisotropic medium, 168-169 for J = I IP and OP pairs, 216, 219 of phonons, 138, 156, 166 renormalized, 181 of vibrons, 64-66 rotational, 88-91 Displacement due to uniform distortion, 175 distribution function, 134, 153, 171 vibrational, 57 Green function, 146 Crystal-field splitting of J = I impurity, operator, 149, 154 191, 193 Displacement coordinates, 133, 140, 146 phonon self-energy contribution to, 186, Displacement correlation matrix, 137, 163, 188 170 static, 176, 183, 185 magnitude ellipsoid of, 164, 168 total, 189, 204 principal axes of, 164 Cumulants, 240 see a/so Correlation function Dissociation energy, 5, 6 ~-effect, 212, 220 Doublet-splitting interaction, 213, 216 Debye frequency, 87, 258 Dunham model, 20-23 Debye model, 170 Dynamical matrix, 135, 146, 152 anisotropic, 156, 168-169, 188 for hcp lattice, 161, 187 Effective interaction for J = I rotons in solid HD, 101 between J = 1 impurities, 198 for J = 2 rotons, 94, 95 between J = 0 impurities, 269, 271 for phonon-mediated interaction, 232 for "spin-lattice" coupling, 178 for phonons, 137, 138, 166,213 Effective-mass approximation, 66 Index 299 Einstein frequency, 133, 170 Equivalent operators, 290 Einstein model, 220 for angular momenta, 244 generalized, 170 for crystal field, 182 Elastic constants, 157 Euler angles, 278 Elastic energy density, 156, 189 between crystal and local frames, 181, Elasticity tensor, 158 243 irreducible parts of. 158-159 Excess binding energy spherical components of. 158 ofJ= 1 impurity, 173, 174, 175 Electric dipole moment of J = 0 impurity in ordered J = energy in field, 38 solids, 246, 249 of isolated molecule, 17 Excess specific heat: lee Specific heat Electric quadrupole-quadrupole inter• action: see EQQ interaction Face-centered structure, 54 Electromagnetic field ordered, 225 around molecule, 9 transition to, 226 correlation function of. 10 unit cell and primitive vectors, 54 of multipole, 37 Fermion operators, 243 of quadrupole, 118, 174 Fixed pairs and triples model, 198 Energy bands: see Librational, Rotational, Fixed pairs model, 198 and Vibrational energy bands Force constants, 134 Enriched ortho-H2 and para-D2 , 225 general, 155 EQQ-cluster model, 201 instantaneous, 143 EQQ coupling constant, 39, 103, 112, 201 phonon renormalized, 143 effective, 203, 210 Free energy difference of fcc and hcp for J = 2 band, 92 structures, 226 phonon renormalized, 214 Free energy of J = I solids, 240 EQQ Hamiltonian in mean-field theory, 238 diagonal and off-diagonal parts, 235 Frustrated alignment, 64, 230: see also De• in J = I solids, 236 gree of frustration in ordered J = in second quantized libron form, 244 I solids in second quantized roton form, 252 EQQ interaction, 37, 39, 92, 103,201 Galileo transformation, 150 eigenstates Gaussian wave function, 133 for arbitrary J, 103 correlated, 136, 142, 164 in semiclassical theory, 104 generalized, 172 for two J = I molecules, 202 uncorrelated, 133 for two J = 2 molecules, 112-113 g-factor expectation value over Neel state, 233 of deuteron, 266 mean-square value for J = I impurities, of proton, 256 264 Gradient formula, 37, 182,213,257, 285 phonon renormalized, 197,212-214 Green function reduced matrix elements of, 202 displacement, 146, 147 reduction by zero-point orientational of vibron, 70-72 motion, 229 walk-counting calculation of, 73-75 relative to local site frames, 242 second-order effects due to, 217 Hamiltonian vibrational off-diagonal part of, 107 harmonic libron, 244 Eq uat ion s-of-motion harmonic oscillator, 148 for elastic deformations, 157 For J = I pairs, 214, 216 for lattice vibrations, 135 in equivalent-operator form, 218 for quantum crystals, 152, 155 of J = 2 rotons in second-quantized Equation-of-state, 45, 57 form, 105 300 Index of rotational motion in the solid, 88 Independent polarizability, 114, 116, 122, "spin-lattice", 178, 181 127, 246 unperturbed rotational, 216 Induced dipole moments, 116-119 see also EQQ Hamiltonian; Hamiltonian additive, 119 matrix; Lattice Hamiltonian; Spin hexadecapole, 119, 127 Hamiltonian isotropic part of, 118 Hamiltonian matrix for pair of J = 1 impurities, 208-209 of J = 1 rotons, 101 quadrupole, 118-119, 208 of J = 2 rotons, 95, 96 spherical components of, 117 Harmonic approximation, 134, 146; see in terms of total angular momentum also Hamiltonian states, 118 Hartree wave function, 132, 133, 141 Induction energy, 36, 37, 176 Heisenberg uncertainty Inflexion point in pair potential, 139 in energy of J = 1 impurity, 260, 262 Infrared spectrum, 122 in J = 0 state, 7 k = 0 selection rule, 123 Heitler-London wave function, 43 phonon branches in, 126 Hexadecapole-hexadecapole interaction, Q,(O) + S,,(O) band, 127 39, 105 S,,(O) line, 123 Hexadecapole moment S,,(O) + S,,(O) band, 125 ab initio values of, 14 S,(O) band, 127 adiabatic, 13 superradiance in, 124-125 adiabatic matrix elements of, 16 time effects in, 273 defimtlon, Ll U transitions, 127 see also Induced dipole moment see also selection rules Hexagonal close-packed structure Inhibition of rotation diffusion, 273 a-planes and ,B-planes, 55, 90 Integrated absorption coefficient in-plane (IP) pairs, 56 of S,,(O) line, 123 inversion symmetry, 55 of S.,(O) + S,,(O) band, 125 nonprimitive vector, 54, 56 Intermolecular potential out-of-plane (OP) pairs, 56 decomposition of, 32 partial reconversion to, 226 definition of, 31 unit cell and primitive vectors, 54 isolated pair, 46 Homogeneous deformation, 156, 175 long-range dispersion, 40 Hooke's law, 157 nonadditive, 36, 226 Hopping matrix elements radial functions of, 44, 49 in J = 1 roton band, 98 short-range, 42 in J = 2 roton band, 92 single-molecule and coupling terms, 33, of S,(O) complexes, 109 34 vibrational, 62 see also Anisotropic interaction; Vibra• Horizontal transitions, 260 tional interaction Homer conditions, 144, 154, 164, 172 Internuclear potential Hybridization of phonons and rotons, 87, anharmonicity in, 67 100, 132 diagonal corrections, 4, 5 effective, 3-5, 20 nonadiabatic effects, 4 Identical particles relativistic effects, 4, 5 indistinguishability of, 23 Intramolecular interaction permutation of, 4 nuclear dipole, 191 Impurity level, 69-72 rotation-vibration, 106 Impurity states, 69-72 Intramolecular potential Incremental polarizability, 122 definition of, 32 Index 301 effective, 57 k = 0 modes, 244 perturbed, 58 mean-field excitation energy of, 248 Irreducible parts virtual, 246 of polarizability tensor, 19 Libron anharmonicity, 242, 244 of second-rank tensor, 282 Local frames, 165,213,231-233,268 Isotope effects, 30 Localized vibrational states, 69-75 Long-range order, 226 Jastrow cut-off function, 143, 171: see lIlso Long-range order parameter Short-range correlations in J = 0 solids, 250 Jump frequency of J = 0 impurities in J = 1 solids, 235 at low temperatures, 272 measurements of, 234 reduced, 271 unperturbed, 270 Magnetic dipole-dipole interaction Jump frequency of J = I impurities. 259 intermolecular, 209, 256-257 average. 262, 263. 265 between total nuclear spins, 257 ratio of J = 0 and J = 1, 272 Magnetic dipole moment reduced, 262 of isolated molecules, 17 unperturbed, 260 nonadiabatic, 18 see also Reduction factor rotational, 17, 258 Jump frequency of J = I pairs, 268 Magnitude ellipsoid, 164, 168, 169 Master equation, 266 Kinetic model, 267, 273 Maximum-I rule, 38, 39 Kirkwood expansion method, 240, 246 Metastable species, 26: see also Ortho and Koster-Slater method, 70 para species Kubical harmonics, 89 Microwave spectrum of J = 1 pairs, 206, 207 Lamb shift, 4, 5, 132 polarization properties, 208-209 Larmor frequency, 191,209 selection rules, 208-209 Lattice distortion, 176, 189,237 Mixed, double excitations, Ill: see also Lattice Hamiltonian Rotation-vibration bands dimensionless form of, 138 Mixed rotation-vibration excitations, 87, effective harmonic, 142, 147, 153 105, 126 harmonic, 134, 136 Molecular field, 237, 247 of nn Einstein oscillators, 220 fluctuations in, 239, 241 quantum crystal, 153 Multipole expansion, 10, 11 Lattice oscillators, 136 Multipole moments Lattice sums adiabatic, 13 crystal-field, 88 adiabatic matrix elements of, 14 J = 2 energy band, 93, 96, 98 of axially symmetric systems, 13 phonon renormalization of, 93, 96 instantaneous components, 13 for random distribution of impurities, nonadiabatic contributions, 16 262 spherical components of, 11 of rotational dipole moments, 123 transformation properties of, 12 Legendre polynomials, 278 Multipole-multipole interactions, 36-39, Lennard-Jones potential, 45, 139 213 Librational energy bands, 242 for axial multi poles, 39 band origin, 244 phonon renormalized, 214 k = 0 states, 244 second-order effects of, 217 .Iee also Librons see also Dipole-dipole interaction: EQQ Librons, 242 interaction harmonic Hamiltonian, 244 Multiple rotational excitations, 102 302 Index Natural line width, 77, 124 Pairs of J = 0 impurities Nearest neighbors of IP and OP pairs, 170 effective interaction of, 250 Neel state, 230, 237 wave function of, 249 NMR spectroscopy Pairs of J = I impurities fast regime in, 192, 211 crystal-field interaction, 216 of HD impurities, 194 difference in properties of IP and OP, of J = I pairs, 209, 211 181,221 of single J = I impurities, 191 direct interaction, 216 Nonadditive perturbations due to impuri• dissociation of, 266 ties, 80 doublet splittings, 214 Nonaxiality parameter, 165, 166, 168-171, effective Hamiltonian, 214-218 179 EQQ eigenstates of, 202 Nonaxially symmetric interaction, 216 formation of, 267 Non-EQQ interactions, 212, 216, 217 inversion symmetry in, 208, 215 Normal-mode coordinates, 135, 136 jump frequencies of, 268 transformation to, 135 splitting of ground state of, 204, 205. Normal modes, 135, 158 211, 214, 268 longitudinal and transverse, 138 width of levels of, 268 Nosanow approximation, 145, 155, 164. 171 Pake doublet, 209, 236 in "spin-lattice" coupling, 178, 182 Parity Nuclear relaxation time, 271 of spherical harmonics, 24 of spin functions, 24 Oblateness parameter 11\.:1, !6/), !68, !70, uf ivial allguiar momentum eIgenfunc• 179 tions, 113 Octupole moment, 12 Partition function, 190 Operator identities, 149 of clusters of J = I impurities, 199 Order-disorder transition, 226, 233, 269 of a pair of J = I molecules, 203 Orientational order parameter: see Long- of pure J = I solids, 240 range order parameter of single J = I impurities, 190 Orientational polarization: see Rotational Pa3 structure, 230, 251 polarization ground-state energy of, 233 Ortho and para species rule for associating body-diagonals to definition, 25 sublattices, 232 equilibrium concentration, 25 Pauli exclusion principle, 4, 30, 256 Orthon, 260 Pauli spin functions, 42 Ortho-para conversion Percolation properties, 76, 85, 235 definition, 26 Phonon branches in infrared spectra, 125 double, 258 Phonon mediated interaction, 197, 222 effect on NMR spectrum, 226 Phonon renormalization, 110 in isolated molecules, 26 static, 173, 239 in larger clusters, 274 see also Renormalized potential due to paramagnetic impurities, 26 Phonons, 136 rate equation for, 258 creation and annihilation operators of, resonant, 256, 258 136 Ortho-para conversion interaction, 257 see also Lattice Hamiltonian bilinear terms in, 258 Phonon self-energy effects: see Self-energy effective resonant, 258-259 in phonon field Overlap effect in conversion, 270, 271 Polarizability Overlap induction mechanism, 117 adiabatic matrix elements of, 18, 19 anisotropy of, 18, 114, 209 Pair correlation function, 154, 213 for clamped nuclei, 18 Index 303 incremental, 122 Quick freezing, 198 isotropic part of, 18, 114, 208 Polarizability tensor Racah bipolar harmonic, 289 angular dependence of Cartesian compo- matrix elements of. 289 nents, 116 Racah coefficients. 285 definition of, 18 Racah spherical harmonics, 278 irreducible components of. 19 matrix elements of. 287 reduced matrix elements of. 206 Radiation damping, 77, 124 rotational matrix elements of. 120 Raman amplitude, 77 transformation of. 115 Raman librational spectrum, 245 Polarization Raman rotation spectrum in MW spectra, 208-209 broadening of 5,,(0) line at ultrahigh in right-angle scattering, 114, 122 pressures, 252 of 5,.(0) infrared line, 123 double and U transitions, 122 Polarization energy intensity ratios in 5,,(0) triplet, 121 of] = 0 impurity in] = 1 solid, 247- of] = 0 solids, 120 249 k = 0 selection rule, 119 of] = 1 impurity in] = 0 solid, 174, of pairs of] = I impurities, 205 175, 176 Raman selection rules: see Selection rules nonadditive term in, 250 Raman vibration spectrum Polarization index anisotropic cross section, 76 of rotons, 87, 92 anomalous ortho-para intensity ratios. of Zeeman excitations, 241 76-81 Polarization vectors, 135, 160 broadening and shifts of Q lines, 81-84 quasi transverse and quasilongitudinaL of cluster, of.l = I molecules, 84 161 of isolated molecules, 75 Potential energy surface, 30 k = 0 selection rule, 77 Preferential adsorption, 26, 225 superradiance in, 77 Pryce perturbation method, 184 at ultrahigh pressures, 83, 252 Random distribution of impurities, 198 Quadrupolarization, 235 Random walk process, 259 Quadrupole bonds, 247 Rate constant Quadrupole field, 37, 118, 124, 174,247 for decay of] = I pairs, 274 Quadrupole glass, 227 experimental values of, 272-274 Quadrupole-induced strain. 219, 220 for rotation diffusion, 263, 266-267 Quadrupole induction mechanism, 117 strain dependence of, 273 Quadrupole moment see also Rotation diffusion ah initio values of, 14 Rate equation adiabatic, 13 for ortho-para conversion. 258 adiabatic matrix elements of, 15 for OP and IP pairs, 274 definition of. 12 Rate of transitions between] = I impurity expectation value in rotational states, levels, 192, 211 103 Rayleigh expansion, 94 Quantum crystals. 131 Reaction channels, 266 definition of, 140 Reciprocal lattice, 97 Quantum diffusion, 255 Reconversion to hcp phase, 226 Quantum law of corresponding states, 138, Reduced matrix elements, 288. 289 141 Reduced molar volume, 139, 140 Quantum mechanical resonance, 203, 229 Reduction factor incomplete suppression of, 230 of jump frequency of] = 1 impurities, Quantum parameter, 138, 140 266,269 304 Index of J = 0 impurities, 270 superposition approximation for, 102 of long-range pair interactions, 45-47 wave equation of J = 2, 92 Renormalized potential wave function of J = 2, 92, % of crystal field, 181 see also Rotational energy bands of harmonic interactions, 180, 213 of linear "spin-lattice" coupling, 178- Scattering cross section, Scattering effi• 179 ciency, 114-115 of nonharmonic and overlap interac• powder average, 205 tions, 180 Schottky anomaly, 190 see also EQQ interaction; MUltipole• Schrodinger equation multipole interactions for clamped nuclei, 2, 4 Residence time, 259, 260 time dependent, 150 Resonance frequency, 70; see also Quan- for two electrons and two nuclei, tum mechanical resonance of vibrational impurity, 150 Retracing of steps, 267 of vibrons, in matrix form, 65 Right-angle scattering, 114-115, 122 Second-order secular equation, 184 Right eigenstates, 149 Second quantization Rigid rotation, 156 of J = 2 rotons, 105, 252 Rotational constant, 20 of lattice vibrations, 136, 181 Rotational energy bands see also Hamiltonian density-of-states in, 93 Selection rules k = 0 levels in, 94, 96, 100 in infrared spectra, 94, 120, 123 single J = 2,87-98 in MW pair spectra, 208, 209 in solid HD, 98-102 in Raman spectra, 77, 94, 116, 120,205 width of, 93, 101, 104 Self-consistent field see also Rotons electron wave functions, 42 Rotational magnetic moment, 17,258 Hartree-Fock,43 Rotational polarization, 110, 175,220 in Pa3 structure, 236 Rotation diffusion Self-consistent phonons, 131, 143, 147, 151 definition of, 255 Self-energy in phonon field, 131, 173, 220 inhibition of, 273 classical theory of, 189 of para-D2 in solid ortho-D2 , 266 difference for IP and OP pairs, 220 see also Rate constant of J = I clusters, 201 Rotation matrices, 279 nonadditivity in, 219 decomposition rule of, 282 of rotating molecule, 185, 187 Rotation operators, 279 see also Crystal-field splitting of J = Rotation - vibration bands, 106-114 impurity density-of-states in, 127 Short-range correlation function, 144, 145, second- and third-order shifts in 5 1(0) 154 level,107-109 Short-range correlations the 51(0) band, 109 in lattice vibrations, 144, 147, 153-154 spectra of, 126 in ordered J = I solids, 239 see also Bound complexes Simultaneous jumps of J = I excitations, Roton-phonon coupling, 100; see also Hy• 263 bridization of phonons and rotons Single-particle distribution functions, 133 Rotons Softening of roton modes, 251 bound states of, 99, 103 Specific heat effective repulsive interaction between, at constant volume, 190 103 of J = I clusters, 200 spin index of, 92 of J = I pairs in solid H2 , 273 Index 305 of single J = 1 impurities, 190 trigonal axis, 89 at small J = 1 concentrations, 204 Symmetry species, 4, 23-25 of solid O2 , 191, 192, 193 even and odd rotational state, 24 Spectroscopic constants ortho and para, 25 definition of, 21 empirical values for isolated molecules, T configuration 22 in infinite 1, 2, and 3 dimensional sys• perturbed values, 59, 88 tems, 228-230 Spectroscopic stability, 78 of isolated J = 1 pairs, 203, 228 Spherical components, 283 Term values of vibrating roton, 21 Spherical harmonics, 277 perturbed, 59 addition theorem, 279 Three-particle distribution function, 171 decomposition rule, 283 Time effects in solid H2 , 255, 272 polarized, 290, 291 diffusion model, 266 transformation under rotations, 278 in infrared spectra, 273 Spin Hamiltonian, 184, 215 kinetic model, 267 "Spin-lattice" coupling, 176 in NMR spectra, 272 dynamic effects due to, 179 in pressure measurements, 273 effective, 177 in specific heat measurements, 273 interaction constants of, 183 Time reversal invariance, 215 isotropic and anisotropic parts, 186 Total angular momentum eigenstates, 202 linear, 183 Transition temperature of order-disorder point symmetry of, 184 transition renormalized, 178, 179 effect of fluctuations, 241 Spontaneous symmetry breaking, 251: .Iet' on fcc lattice, 226 a/so Broken symmetry in mean-field theory, 238 Static phonon renormalization: see Phonon Triangle condition, 280 renormalization Triangles in hcp lattice, 198, 199 Statistical weight Triple clusters, 198 of distribution of impurities over lattice, Tumbling motion of J = I impurities, 264: 199 see a/so Correlation time of J = I of rotational levels, 25 impurities Stimulated emission processes, 119 Two-particle distribution functions, 134, Stochastic perturbation theory, 261 163-171 Strain tensor, 156 Stress tensor, 157 U transitions, 127 Structural phase change, 225, 226 Sublattices, 226, 230-231 Van der Waals complexes, 205 Substitutional impurities, 173 Van der Waals contribution of "4,216 Superiattice of J = 1 impurities, 81, 265 Velocity of propagation of phonons in hcp Superposition approximation for rotons, lattice, 161 102, 105 Vibrational energy bands, 57, 61 Superradiance, 77 coupling constant, 64 of 5,,(0) transition, 124-125 in hcp lattice, 65 Symmetry axis of molecule, 230 k = 0 levels, 65 Symmetry breaking: see Broken symmetry shrinkage due to impurities, 82 Symmetry properties Vibrational impurity, 68 under permutations of identical particles, Vibrational interaction 4 coupling terms, 33, 57, 61 site (point) symmetry, 88 single-molecule term, 57 306 Index Vibrational levels of isolated molecules, 6 Width of J = 1 pair levels, 268; see also Vibron,63 Raman rotation spectrum; Raman effective mass of, 66 vibration spectrum Green function of, 72 Wigner-Eckart theorem, 288 kinetic energy operator of, 72 Wigner 3-j, 6-j, and 9-j symbols, 281, 285, wave function of, 63 287 Virtual phonons, 110, 176,222 Voigt indices, 157 Zeeman excitations, 241 creation and annihilation operators of, 243 Walk-counting method excitation energy of, 241, 248 vibrational energy band, 65 polarization index of, 243 vibrational impurity problem, 73 Zero-phonon features in infrared spectra, Wave equation: see Schrodinger equation 117 Wave functions Zero-point distributions, 141 adiabatic, 3 Zero-point lattice vibrations, 131, 139 phases of, 14 anisotropy in, 175 of rotation-vibration states, 8 average over, 163 of rotons, 92, % Zero-point librational effect, 230, 235, 242 of vibrons, 63 Zero-point orientational motion, 103, 229